A002537 a(2n) = a(2n-1) + 3a(2n-2), a(2n+1) = 2a(2n) + 3a(2n-1).

1, 1, 4, 11, 23, 79, 148, 533, 977, 3553, 6484, 23627, 43079, 157039, 286276, 1043669, 1902497, 6936001, 12643492, 46094987, 84025463, 306335887, 558412276, 2035832213, 3711069041, 13529634721, 24662841844, 89914587851

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.

Links

• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
• Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy]
• Index entries for linear recurrences with constant coefficients, signature (0, 8, 0, -9).

Programs

Maple

A002537:=(1+z-4*z**2+3*z**3)/(1-8*z**2+9*z**4); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

LinearRecurrence[{0,8,0,-9},{1,1,4,11},40] (* Harvey P. Dale, Jul 24 2012 *)

Formula

a(n)=8a(n-2)-9a(n-4). - Mario Catalani (mario.catalani(AT)unito.it), Apr 24 2003

G.f.: (1+x-4x^2+3x^3)/(1-8x^2+9x^4). a(n)/A002536(n) converges to sqrt(7). - Mario Catalani (mario.catalani(AT)unito.it), Apr 24 2003

a(n+1) = x^n + (-1)^n*(x-2)^n where x = (1+sqrt(7)) and the term is divided by 2 for a(2) and a(3), 4 for a(4) and a(5)... 2^n for a(2n) and a(2n+1). - Ben Paul Thurston, Aug 30 2006

Extensions

More terms from James Sellers, Sep 08 2000